Method for pressing liquid out of fibrous or woodlike material

ABSTRACT

A method of pressing liquid out of fibrous material, such as particulated sugarcane or the like, in which the material is subjected in a tapering screw press to a continuously increasing pressure as it is forced from the large to the small diameter and of the press so that the material will leave the small diameter end of the press in substantially dry condition while the liquid pressed out from the material is caused to flow in countercurrent to the material and is discharged from the press in the region of the large diameter thereof.

United States Patent 808,193 12/1905 Bussells Kurt Heinrich An Mevissen 5, Wevelinghoven, Germany 880,232

Nov. 26, 1969 Division of Ser. .\'0. 745,055. A pr. 2. 1969. July 27, 197 l Inventor Appl. No. Filed Patented METHOD FOR PRESSING LIQUID our or FIBROUS 0R woonuxn MATERIAL 6 Claims, 5 Drawing Figs.

U.S.C| 100/37, 100/72, 100/117, 100/148 Int. Cl ..A47j 19/02, B30b 9/02, B30b 9/12 FieldofSearch [00/117, 110, 111, 145-168, 150,71-74, 39,48

References Cited UNITED STATES PATENTS oonnx 1,191,760 7/1916 Canfield 100/145 X 1,653,831 12/1927 Archidiacano... 100/145 2,166,547 7/1939 Horath 100/1 17 X 2,355,091 8/1944 McDonaldm, 100/73 X 2,513,974 7/1950 Thomas 100/145 X 3,076,610 2/1963 Rosenfeld et al. 100/117 X 3,230,865 l/l966 Hibbel et a1. l00/l17 X 3,472,159 10/1969 Washburn 100/72 X Primary Examiner-Peter Feldman Attorney-Michael S. Striker ABSTRACT: A method of pressing liquid out of fibrous material, such as particulated sugarcane or the like, in which the material is subjected in a tapering screw press to a continuously increasing pressure as it is forced from the large to the small diameter and of the press so that the material will leave the small diameter end of the press in substantially dry condition while the liquid pressed out from the material is caused to flow in countercurrent to the material and is discharged from the press in the region of the large diameter thereof.

PATENTED JUL 2 7 an SHEET 1 OF 2 METHOD FOR PRESSING LIQUID OUT OF FIBROUS OR WOODLIKE MATERIAL CROSS-REFERENCE TO RELATED APPLICATION The present invention is a divisional application of the copending application, filed Apr. 2, 1969, under the Ser. No. 745,055, and entitled Screw Press For Pressing Liquid Out Of Fibrous Or woodlike Material.

BACKGROUND OF THE INVENTION The present invention relates to a method for pressing liquid out of fibrous or woodlike material such as particulated sugarcane, crushed sugarcane, fruits, for instance pineapples, grapes or olives and also leaves, grasses, wood flower or the like.

Screw presses in which the material is pressed against, and the pressed out liquid discharged through a sieve are not suitable for treating such fibrous or woodlike material, since the pressure required to press liquid out of such material is so great that a sieve subjected to such pressure would not stand up. In addition, the openings of thesieve would be clogged. Screw presses are known which are able to withstand the necessary high pressure and in which the worm screw cooperates with an uninterrupted surface of the housing. The fluid pressed out of the material flows thereby in countercurrent to the direction in which the material passes through the press. Worm screw presses are also known in which the worm screw tapers from the inlet to the outlet end and is surrounded by a frustoconical surface of the housing and in which the material is subjected to a gradually increasing pressure as the material is forced from the inlet to the outlet end of the press. The inner surface of the housing is thereby provided with axially extending grooves in order to prevent rotation of the material together with the screw.- The disadvantages of all these known screw presses is that the space defined between the threads of the worm screw is substantially trapezoidal and that the surface of the worm screw contacted by the material is considerably greater than the inner surface of the housing which is contacted by the material. Therefore, and especially since the material will be compressed between the opposite pressuresurfaces of adjacent screw threads, the friction of the material on the surfaces of the screw will be greater than the friction on the inner surface of the housing, even if the latter is provided with grooves, so that the material will rotate with the screw and not be forced in axial direction of the latter. The material compressed in the grooves of the inner surface of the housing will thereby simply be sheared from the remainder of the material.

It is an object of the present invention to provide for a method in which the material to be treated is continuously forced under quickly increasing pressure from the inlet end to the outlet end of a press and to permit the liquid pressed out I from the material to flow in countercurrent to the latter.

SUMMARY OF THE INVENTION With these objects in view, the method of the present invention of pressing liquid out of fibrous material, such as particulated sugarcane and the like, mainly comprises the steps of continuously forcing the fibrous material in one direction from one to the opposite end of a pressure chamber the cross section of which continuously decreases in said one direction so as to subject the material during its movement in the one direction through the pressure chamber to continuously increasing pressure so as to press liquid contained in the fibrous material out of the latter, discharging the material in substantially dry condition through the other end of the pressure chamber, and causing the liquid pressed out from the material to'flow indirection opposite to the'one'direction.

The present invention may also include the step of feeding water into the pressed out substantially dry material in such amounts to transform the substantially dry material into a sludge and subjecting the sludge to a second pressing operation.

The pressure chamber may be formed in a tapering screw press and the material is subjected to the continuously increasing pressure by forcing the material from the large to the small diameter end of the press.

The present invention may also include the step of preventing rotation of the material with the worm screw of the tapering screw press.

The method of the present invention may also include the step of throttling the outflow of the material through the small diameter end of the press to a variable degree to thereby maintain the dryness of the material in the region of the small diameter end substantially constant.

The screw press for carrying out the present invention mainly comprises housing means having a first portion provided with an inlet for the material and an outlet for the liquid pressed out of the material and a pressure portion having an inner frustoconical surface communicating at the large diameter end with the interior of the first housing portion and having an outlet end for the material at its small diameter end, and worm screw means rotatably arranged in said housing means, the worm screw means having a frustoconical portion in the pressure portion of the housing means and having screw convolutions which extend in direction towards the small diameter end of the worm screw means substantially parallel to the axis thereof and ending at the small diameter end of the screw means in a cylindrical portion coaxial with the outlet end and forming with the inner surface of the latter an annular outlet opening, the outer edges of the convolutions being spaced a small distance from the inner frustoconical surface of the housing means.

The outlet end of the housing gradually increases in cross section to form with the aforementioned cylindrical portion of the worm screw means an annular opening gradually increasing in a nozzlelike manner. The screw convolutions of the frustoconical portion of the worm screw means are formed by a spiral base surface extending substantially parallel to the axis of the worm screw means and a spirally wound pressure shoulder including an obtuse angle with the base surface and joined by a filler thereto.

In this construction the space between the worm screw and the inner surface of the housing is defined only by three faces, that is the face of the spirally wound pressure shoulder, the spirally wound base face of the screw, and the inner frustoconical surface of the housing. In this arrangement the advance of the material to be pressed is produced by the pressure shoulder of the screw. The friction between the material and the smooth surfaces of the screw is thereby smaller than the friction of the material on the inner frustoconical surface of the housing, especially if the latter is provided with grooves. The material forced by the worm screw from the inlet to the outlet end of the press will be compacted in the triangular space defined between the screw faces and the inner frustoconical surface of the press against the pressure shoulder of the screw so that the liquid pressed out from the material will, due to the pressure which increases steeply toward the outlet end of the press, find a spirally wound path through the apex of the space in which a smaller concentration of the solid material will occur so that the liquid will flow in countercurrent to the solid material toward the inlet end of the press. In a screw press according to the present invention the triangular cross section of the aforementioned space will progressively become smaller from the inlet to the outlet end of the press, even if the helix angle of the thread of the worm screw is maintained constant. The nozzlelike expanding annular opening at the outlet end of the housing will result in a sudden expansion of the extruded solid material at the outlet end.

The cone angle of the conical portion of the worm screw according to the present invention is between 40 and preferably substantially 54.

Preferably, the worm screw means has in the first portion of the housing a substantially cylindrical feed screw portion with an inclined pressure surface, beneath which a semicylindrical sieve is arranged in the first portion of the housing through which the liquid pressed out ofthe material may pass.

To improve the efficiency of the screw press used for carrying out the method of the present invention, the helix angle of the pressure shoulder of the frustoconical portion of the worm screw means preferably increases towards the outlet end of the press. Since the friction angle or angle of repose of the material to be pressed increases as the material becomes drier, the increasing helix angle will improve the efficiency of the press. Preferably, the inner frustoconical surface of the press housing is provided with a plurality of grooves extending spirally wound in direction of rotation of the worm screw means from the large towards the small diameter end of the surface. Thereby, the length of the grooves is considerably increased and feeding of the material towards the outlet end of the press assured. In order to reduce the danger of shearing off the material in the grooves by the rotating pressure shoulder of the worm screw, the helix angle of the grooves is preferably held at an angle substantially 90 minus the sum of the helix angle of the shoulder face of the worm screw means and the friction angle of the material on the surface of the worm screw means.

Preferably, each of the grooves has, in a cross section transverse to the axis of the inner frustoconical surface, a surface portion extending in direction of rotation of the worm screw means substantially tangential to the inner surface of the housing means and a packing surface portion substantially normal thereto joined to the tangential surface portion by a fillet and by a small radius to the inner frustoconical surface. The packing surface will be effective to prevent rotation of the material with the worm screw means, while the rounded transition between the packing surface and the inner surface of the housing will prevent shearing of the material.

According to a further feature of the present invention, the press may include also an annular member fixed in the housing between the cylindrical feed screw portion and the frustoconical portion of the worm screw means and a plurality of guide vanes fixed to the annular member, wherein the guide vanes are inclined in direction of rotation of the worm screw means relative to the axis of the latter and project from the annular member nearly up to the cylindrical core of the feed screw portion. Preferably, the guide vanes are arranged substantially at right angles to the screw thread of the feed screw portion. This arrangement will assure that the material fed into the inlet end of the press will be pressed automatically into the spaces between the guide vanes and thus be fed into the conical portion of the worm screw means. At the same time, the material located at any time between the guide vanes will form a filter for the liquid flowing in reverse direction and this filter will retain small particles carried by the liquid.

In order to assure that during changing output of the press, a substantially constant dryness of the pressed out material will be maintained, the press for carrying out the method according to the present invention, may also include a throttling member axially adjustably mounted on the cylindrical portion at the small diameter end of the worm screw means and having an annular face facing the outlet end of the housing and defining with an end face of the latter an annular space gradually increasing in radially outward direction. The position of the throttling member is preferably hydraulically adjustable to change thereby the cross section of the annular space through which the material is discharged so that by changing this cross section in accordance with the output of the press, the desired dryness of the pressed material may be sustained substantially constant.

In order to extract extractive matter from the pressed out material, liquid, for instance water, is fed according to a method of the present invention under pressure into the pressed out material in the region of the smallest noule diameter at the outlet end of the press so that the liquid will penetrate into the expanding material and the thus treated material is then subjected to a second pressing operation.

The liquid thus penetrates into the rapidly expanding material before air can penetrate thereinto. Calculations have shown that the dry content of the pressed out crushed sugarcane is at least 62 percent after each pressing operation, whereby a sugar extraction of 97 percent may be obtained.

The novel features which are considered as characteristic for the invention are set forth in particular in the appended claims. The invention itself, however, both as to its construction and its method of operation, together with additional objects and advantages thereof, will be best understood from the following description of specific embodiments when read in connection with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWING FIG. 1 is a cross section through the housing of the press according to the present invention and showing the worm screw means in the housing in a side view;

FIG. 2 is a perspective view of a cross section through the housing with the worm screw removed therefrom;

FIG. 3 is a partial cross section taken along the line AA of FIG. 2;

FIG. 4 is a side view of a double press according to the present invention, with some of the parts shown in cross section; and

FIG. 5 is a partial cross section taken along the line BB of FIG. 4.

DESCRIPTION OF THE PREFERRED EMBODIMENTS Referring now to the drawings, and more specifically to FIGS. 1 and 2 of the same, it will be seen that the screw press of the present invention comprises housing means having a first portion 1 open at the top to provide an inlet for the material to be pressed, and a pressure portion 2 having an inner frustoconical surface 3 of a cone angle of substantially 54. The lower part of the interior of the first portion 1 of the housing means is defined by a semicylindrical sieve 4, beneath which the housing portion 1 is constructed as a discharge funnel la for the liquid pressed out of the material. An annular member 5 is fixedly arranged in the housing means between the portions 1 and 2 thereof, and the interior diameter of the annular member 5 is equal to the large diameter end of the frustoconical inner surface 3. A plurality of guide vanes 6 arranged inclined to the axis of the frustoconical inner surface 3 project radially inwardly from the inner surface of the annular member 5. I

A worm screw means is rotatably arranged in the housing portions 1 and 2 and the worm screw means have a screw core 7 which in the region of the first housing portion 1 and of the annular member 5 is substantially cylindrical and which forms in the housing portion 2 a spirally wound base surface 8 substantially parallel to the axis of the core, the diameter as well as the width of which decreases in direction toward the outlet end of the press. In the region of the housing part 1, the core 7 is provided with a spirally wound tooth or flange 9 having an inclined pressure face. The flange 9 is interrupted in the region of the guide vanes 6, which reach nearly up to the outer cylindrical surface of the core, and in the region of the inner frustoconical surface of the housing portion 2, the flange 9 blends gradually into the base face 8 and merges with the latter at 10. From this point on, the base face 8 parallels to the axis forms in the worm screw body 7 a spirally wound pressure shoulder 11 connected to the respective base face portion 8 by a fillet and including with the base face portion an angle greater than preferably The helix angle of the spirally wound pressure shoulder 11 is about 8. Contrary to known worm screws it is however advantageous to increase the helix angle toward the outlet end of the press, and the helix angle may begin at the point 10 with 4 and gradually increases to a maximum helix angle of 9. At the outlet end of the press the pressure shoulder 11 blends into a cylindrical portion portion 12 coaxial with the axis of the worm screw means. The outer edge 13 formed by the base surface 8 and the pressure shoulder 11 is chamfered according to the cone angle of the inner frustoconical surface 3 of the housing portion 2 and is spaced from the latter only through a very small clearance. The base surface 8 forms together with the pressure shoulder 11 and the inner frustoconical surface 3 a spirally wound pressure space 14 which decreases continuously in volume towards the outlet end of the press and through which the material to be pressed is advanced with progressively increasing pressure towards the outlet end of the press.

The outlet opening of the housing portion 2 forms together with the cylindrical portion 12 of the worm screw means 7 at 15 an annular, nozzlelike expanding space. A throttle member 16, arranged axially adjustable on the cylindrical end 12 of the worm screw means cooperates with the nozzlelike end 15. The face of the throttling member 16 which faces the outlet end 15 is formed in such a manner to form with an end face of the housing portion 2 an annular space which gradually increases in radially outward direction.

As best shown in FIG. 2, the inner frustoconical surface 3 of the housing portion 2 is formed with a plurality of grooves 17 which extend spirally wound in direction of rotation of the worm screw means from the large toward the small diameter end of the surface 3. Each of the grooves 17 has in a cross section normal to the axis of the frustoconical surface 3, as shown in FIG. 3, a surface portion I7a extending in direction of rotation of the worm screw means substantially tangential to the inner surface 3 and a packing surface portion 17b substantially normal thereto and joined to the tangential surface portion by a fillet and by a small radius to the inner frustoconical surface 3.

The above-described construction of the worm screw means 7 in the region of the pressure portion 2 of the housing means constitutes an essential feature of the present invention.

In contradistinction to known screw presses in which the space between the screw threads or teeth of the worm screw has a substantially trapezoidal cross section, the spirally wound pressure space 14 in the screw construction according to the present invention has a substantially triangular cross section which decreases continuously towards the outlet end of the press. The spirally wound pressure shoulder 11 serves thereby to produce the necessary pressure and to force the pressed material towards the outlet end of the press, whereas in presses known in the art, each pressure surface of the screw is opposed by a counterpressure surface. Due to the omission of the counterpressure surface it is possible to produce with the press according to the present invention considerably higher pressures than can be obtained with known presses of this type. Experiments have shown that the pressure in a press according to the present invention increases progressively from a pressure of about a half atmosphere at the inlet end of the press to a pressure of 250 to 700 atmospheres above atmospheric pressure at the outlet end of the press, depending on the material to be pressed. Due to the continuously decreasing cross section of the pressure space 14, the material to be compressed is compacted much tighter at the pressure shoulder 11 than at the apex of the pressure space which is directed towards the outlet end. In this way a spirally wound path devoid of tightly compressed material is formed along the apex of the pressure space through which the liquid pressed out of the material may flow in countercurrent to the com pressed solid material back to the inlet end of the press.

The material to be treated fed into the open inlet end of the housing portion 1 is engaged therein by the spirally wound tooth 9 of the worm screw means and pressed through the spaces between the guide vanes 6 into the spirally wound pressure space 14. The pressure shoulder ll advances thereby the material in direction toward the outlet end 15, while subjecting the material to a rapidly increasing pressure. The liquid pressed out from the material flows back to the housing portion I and passes thereby through the compressed material in the annular member 5 to be filtered by this compressed material and through the sieve 4 into the discharge funnel In of the housing portion 1.

The pressed out material leaves the press at 15 where the compressed material expands. The grooves 17 prevent the material from rotating together with the worm screw means, while the grooves, due to their spiral configuration, improve the advancing action produced by the worm screw means.

The worm screw means is driven by drive means not shown in the drawing, and preferably arranged at the left side, as viewed in FIG. I, of the worm screw means.

The axial thrust on the worm screw means is in a press as described, especially if the press is built for a high output, considerable, and it is therefore advantageous to combine two presses in a mirror symmetrical arrangement so that the axial thrusts thereof, acting in opposite directions, will compensate each other. Such an arrangement is shown in FIG. 4.

As shown in FIG. 4, a drive housing 18 is arranged between the two inlet housing portions 1 of the two presses. A drive shaft 19 which connects the two worm screw means 7 is arranged in the drive housing and mounted for rotation on a pair of roller bearings 20 constructed to take up any difference in the axial thrusts of the two worm screw means. A sprocket wheel 21 is keyed to the shaft 19 between the roller bearings 20 and the sprocket wheel 21 is driven by a sprocket chain not shown in the drawing from a drive motor, likewise not shown.

In the embodiment illustrated, each of the pressure shoulders 11 has five convolutions. With a press of such a construction it is p ossible to treat particulated sugarcane in each of the pressure portions 2 of the two presses to obtain at the outlet ends thereof a percentage of solids of 6267 percent. In the operation of the press, it is often desirable to transform the substantially dry material emanating from the press into sludge by feeding water thereinto and for this purpose an annular passage 22 is provided in the region of the outlet end of the press shown at the right side in FIG. 4, and a radial passage 24 communicates at the inner end thereof with the annular passage 22, whereas the outer end of the passage 24 is adapted to be connected to a supply of water under pressure, now shown in the drawing. A plurality of channels 23 communicating with the annular passage 22 open at their inner ends at the outlet end of the press so that water under pressure may be fed into the material as it passes through the smallest diameter of the outlet opening whereby the water, as the material subsequently expands, will penetrate therethrough. The cylindrical end portion 12 of the worm screw means 7 is axially movably mounted, in a manner not shown in the drawing, in a bearing housing 25 which in turn is likewise axially movable fluidtightly guided in a bearing bracket 26 connected to the housing portion 2. The bearing bracket 26 forms a cavity 27 through which pressure oil may be fed through the inlet opening 28 in order to axially move the bearing 25 on the cylindr cal portion 12. The throttling member 16 is rigidly connected to the bearing housing 25 so that during axial adjustment of the latter the cross section of the nozzlelike increasing outlet space formed between one face of the throttling member I6 and the end face of the housing may be adjusted.

If the press according to the present invention is used for pressing juice of particulated sugarcane, then it is possible for a small daily production, to operate the press as shown in FIG. 4 in such a manner that the particulated sugarcane is subjected for instance in the pressure housing portion 2, shown at the right side of FIG. 4, to a first pressing operation in which most of the juice is pressed from the'particulated sugarcane and in which water is fed into the pressed out material to transform the latter into sludge which may be pumped. Since the water will penetrate into the finest pores of the pressed out material during expansion of the latter at the outlet end of the press, nearly all in the pressed out material remaining sugar will be extracted. The sludge is then fed into the inlet end of the left press, as shown in FIG. 4, and in this left press the sludge is subjected to a subsequent pressing operation whereby a total sugar extraction of 97 to 98 percent will be obtained.

On the other hand, if a greater daily production is required, both pressure housing portions 2 of the double press are preferably provided with means for feeding water into the outlet end of the respective press and both presses are operated in the same manner, that is the particulated sugarcane is first subjected in each of the presses to a first pressing operation and the sludge is recirculated into the respective press and subjected to a second pressing operation. Instead of feeding the water through passages in the press housing into the pressed out material, it is also possible to arrange similar water feed passages in the throttle member 16.

It will be understood that each of the elements described above, or two or more together, may also find a useful application in other types of screw presses differing from the types described above.

While the invention has been illustrated and described as embodied in a screw press in which the worm screw is constructed in such a manner that the faces thereof form together with a frustoconical inner surface of the housing a spirally wound pressure space of triangular configuration decreasing in cross section towards the outlet end of the press, it is not intended to be limited to the details shown, since various modifications and structural changes may be made without departing in any way from the spirit ofthe present invention.

Without further analysis, the foregoing will so fully reveal the gist of the present invention that others can by applying current knowledge readily adapt it for various applications without omitting features that, from the standpoint of prior art, fairly constitute essential characteristics of the generic or specific aspects of this invention and, therefore, such adaptations should and are intended to be comprehended within the meaning and range of equivalence of the following claims.

What I claim as new and desire to be protected by Letters Patent is set forth in the appended claims.

1. A method of pressing liquid out of fibrous material, such as particulated sugarcane and the like, comprising the steps of continuously forcing the material in one direction from the large diameter end of a pressure chamber, formed between the frustoconical inner surface of a press housing and a frustoconical worm screw, to the small diameter end thereof by rotating the worm screw while confining the material in a direction transverse to said one direction so as to subject the material during movement in said one direction to continuously increasing pressure to press liquid contained in the fibrous material out of the latter; preventing rotation of the material with the worm screw; discharging the material in substantially dry condition from said small diameter end of said pressure chamber; causing the liquid pressed out from the material to flow within said pressure chamber in a direction opposite to said one direction; and discharging all ofthe liquid pressed out from the material upstream of said large diameter end of said pressure chamber.

2. A method as defined in claim 1, wherein the material is subjected at said other end of said pressure chamber to a maximum pressure ofabout 250-700 atmospheres.

3. A method as defined in claim 1, and including the step of feeding water into the pressed out substantially dry material in such amounts to transform the substantially dry material into sludge, and subjecting the sludge to a second pressing operation.

4. A method as defined in claim 1, and including the steps of feeding water under pressure into the pressed out material in the region of the small diameter end of said pressure chamber in such amounts to transform the material into sludge, and subjecting said sludge to a second pressing operation in said press.

5. A method as defined in claim 1, and including the step of maintaining the dryness of the material in the region of said small diameter end of the pressure chamber substantially constant.

6. A method as defined in claim 1, and including the step of throttling the outflow of the material through said small diameter end of said pressure chamber to a variable de ree to maintain the dryness of the material in the region of sai small diameter end substantially constant. 

1. A method of pressing liquid out of fibrous material, such as particulated sugarcane and the like, comprising the steps of continuously forcing the material in one direction from the large diameter end of a pressure chamber, formed between the frustoconical inner surface of a press housing and a frustoconical worm screw, to the small diameter end thereof by rotating the worm screw while confining the material in a direction transverse to said one direction so as to subject the material during movement in said one direction to continuously increasing pressure to press liquid contained in the fibrous material out of the latter; preventing rotation of the material with the worm screw; discharging the material in substantially dry condition from said small diameter end of said pressure chamber; causing the liquid pressed out from the material to flow within said pressure chamber in a direction opposite to said one direction; and discharging all of the liquid pressed out from the material upstream of said large diameter end of said pressure chamber.
 2. A method as defined in claim 1, wherein the material is subjected at said other end of said pressure chamber to a maximum pressure of about 250-700 atmospheres.
 3. A method as defined in claim 1, and including the step of feeding water into the pressed out substantially dry material in such amounts to transform the substantially dry material into sludge, and subjecting the sludge to a second pressing operation.
 4. A method as defined in claim 1, and including the steps of feeding water under pressure into the pressed out material in the region of the small diameter end of said pressure chamber in such amounts to transform the material into sludge, and subjecting said sludge to a second pressing operation in said press.
 5. A method as defined in claim 1, and including the step of maintaining the dryness of the material in the region of said small diameter end of the pressure chamber substantially constant.
 6. A method as defined in claim 1, and including the step of throttling the outflow of the material through said small diameter end of said pressure chamber to a variable degree to maintain the dryness of the material in the region of said small diameter end substantially constant. 